TRANSVERSE VIBRATIONS OF NONHOMOGENEOUS RECTANGULAR PLATES OF UNIFORM THICKNESS USING BOUNDARY...

Int. J. of Appl. Math and Mech. 6 (14): 93-109, 2010.

TRANSVERSE VIBRATIONS OF NONHOMOGENEOUS

RECTANGULAR PLATES OF UNIFORM THICKNESS USING

BOUNDARY CHARACTERISTIC ORTHOGONAL POLYNOMIALS

R. Lal1, Y. Kumar

1, and U.S. Gupta

2

1Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee – 247 667,

India.

Email: [email protected] 2Department of Applied Sciences, Academy of Business and Engineering Sciences,

Noida – 201 305, India.

Received 7 January 2009; accepted 4 October 2009

ABSTRACT

Analysis and numerical results for the free transverse vibrations of uniform nonhomogeneous

rectangular plates have been presented using boundary characteristic orthogonal polynomials

in the Rayleigh-Ritz method on the basis of classical plate theory for four different

combinations of clamped, simply supported and free edges. Gram-Schmidt process has been

used to generate these orthogonal polynomials satisfying essential boundary conditions. The

nonhomogeneity of the plate is assumed to arise due to linear variations in Young’s modulus

and density of the plate material with the space coordinates. Effect of the nonhomogeneity

with varying values of aspect ratio on natural frequencies is illustrated for the first three

modes of vibration. Three dimensional mode shapes have been presented for all the four

boundary conditions. A comparison of results with those available in the literature shows a

close agreement.

Keywords: Nonhomogeneous, rectangular, uniform thickness, Rayleigh-Ritz.

1 INTRODUCTION

Plates of various geometries are key components in many structural and machinery

applications. Their design requires an accurate determination of their natural frequencies and

mode shapes. Although, a vast literature for the flexural vibrations of isotropic rectangular

plates of uniform thickness exists and reported in references (Leissa 1969; Leissa 1973;

Gorman 1982; Timoshenko and Woinowsky 1984; Shames and Dym 1985; Inman 1994; Rao

2004; Reddy 2007), to mention a few, even the study of their vibrational behavior is

continuing till today. In a paper, Zitnan (Zitnan 1999) analyzed the transverse vibrations of

rectangular and skew plates by Rayleigh-Ritz method using B-spline functions. Gorman

(Gorman 2000) extended the superposition-Galerkin method for analyzing the free vibration

of completely rectangular plates. Arenas (Arenas 2003) used the virtual work principle in

obtaining the natural frequencies for clamped rectangular plates of arbitrary aspect ratio and

an estimate for the modal density. Chen et al. (Chen et al. 2004) proposed a meshless method

for solving the eigenfrequencies of clamped circular and rectangular plates using the radial

94 R. Lal et al.


basis functions. Very recently, Yeh et al. (Yeh et al. 2006) analyzed the free vibrations of

clamped and simply supported rectangular thin plates using finite difference and differential

transformation method. Kerboua et al. (Kerboua et al. 2007) presented a semi-analytical

approach namely, hybrid method for vibration analysis of rectangular plates. Li et al. (Li et al.

2008) developed an analytical method for the vibration analysis of rectangular plates with

elastically restrained edges.

Nonhomogeneous elastic plates have their applications in the design of space vehicle, modern

missile and aircraft wings (Chakraverty and Petyt 1999). Various models for the non-

homogeneity of the plate material have been assumed by the researchers and a brief review is

given in a recent paper by Lal and Dhanpati (Lal and Dhanpati 2007). Bose (Bose 1967)

analyzed the vibrations of thin non-homogeneous circular plates with a central hole assuming

that the Young’s modulus rEE 0 and density r0 of the plate material vary with the

radius vector r where 00 ,E are constants. Biswas (Biswas 1969) considered a non-

homogeneous material for which rigidity ze 1

0

and density z

e 1

0

both vary

exponentially where 00 , are constants. Rao et al. (Rao et al. 1974) studied the vibration of

non-homogeneous isotropic thin plates with Young’s modulus and density given by

)1(0 xEE and )1(0 x . In a series of papers, Tomar et al. (Tomar et al. 1982;

Tomar et al. 1982; Tomar et al. 1983; Tomar et al. 1984) assumed exponential variation of

Young’s modulus and density i.e. xeEE ),(),( 00 in the dynamic behavior of non-

homogeneous isotropic plates of variable thickness of different geometries. Elishakoff

(Elishakoff 2000; Elishakoff 2000) obtained closed form solutions for the axisymmetric

vibrations of isotropic inhomogeneous clamped and free circular plates. Recently, a number

of papers have appeared in the literature analyzing the effect of nonhomogeneity on the

vibrational behavior of elastic plates. Lal et al. (Lal et al. 2004) studied axisymmetric

vibrations of non-homogeneous polar orthotropic annular plates of variable thickness in

which non-homogeneity and density of the plate material vary in radial direction i.e. xx

r eEEeEE

21 , and xe 0 . Gupta et al. (Gupta et al. 2006) have analyzed free

axisymmetric vibrations of non-homogeneous isotropic circular plates of quadratically

varying thickness where Young’s modulus and density are assumed to vary exponentially in

one direction i.e. xeEE 0 and xe 0 , . Chakraverty et al. (Chakraverty et al.

2007) assumed the quadratic variation of Young’s modulus and density in one direction to

study the effect of non-homogeneity on natural frequencies of elliptic plates. Lal et al. (Lal et

al. 2008) studied buckling and vibration of non-homogeneous x

y

x

x eEEeEE 21 ,( and

)0

xe orthotropic rectangular plates of varying thickness under biaxial compression. In

a recent survey of literature, the authors have not come across any paper in which the

nonhomogeneity of the plate material depends upon both the space coordinates.

The present paper deals with the free transverse vibrations of thin nonhomogeneous

rectangular plates of uniform thickness using two dimensional boundary characteristic

orthogonal polynomials in the Rayleigh-Ritz method on the basis of classical plate theory.

Nonhomogeneity of the plate is assumed to arise due to linear variations in Young’s modulus

and density of the plate material with the space coordinates. The value of Poisson ratio is

assumed to remain constant. The effect of nonhomogeneity on the vibrational characteristics

of rectangular plates with various values of aspect ratio has been studied for four different

combinations of boundary conditions.

Transverse Vibrations of Nonhomogeneous Rectangular Plates 95


2 FORMULATION OF THE PROBLEM

Consider an isotropic nonhomogeneous rectangular plate of uniform thickness h with the

domain byax 0,0 in xy plane, where a and b are the length and the breadth of

the plate, respectively. The x and y axes are taken along the edges of the plate and axis of

z is perpendicular to the xy plane. The middle surface being 0z and origin is at one of

the corners of the plate as shown in figure 1(a).

(a) (b)

Figure 1: (a) Geometry of the plate (b) boundary conditions.

The expressions for strain energy and kinetic energy of the plate with flexural rigidity

)1(12/ 23 EhD , Young’s modulus ),( yxE , density ),( yx and Poisson’s ratio , are

given by

a b

xyyyyyxxxx dydxwwwwwDV0 0

222])1(22[

2

1 (1)

a b

t dydxwh

T0 0

2

2 , (2)

For harmonic solution, the deflection function ),,( tyxw is assumed to be

)sin(),(),,( tyxWtyxw , (3)

where ),( yxW represents the maximum transverse displacement at point ),( yx , the

circular frequency and t is the time.

Substituting relation (3) in expressions (1) and (2), the expressions for maximum strain

energy and kinetic energy of the plate become

a b

xyyyyyxxxx dydxWWWWWDV0 0

222

max ])1(22[2

1 (4)

96 R. Lal et al.


a b

dydxWh

T0 0

22

max2

. (5)

According to the Raleigh-Ritz method equating equations (4) and (5), one gets the Rayleigh

quotient as

a b

a b

xyyyyyxxxx

dydxWh

dydxWWWWWD

0 0

2

0 0

222

2

])1(22[

. (6)

Introducing nondimensional variables ahHaWWbyYaxX /,/,/,/ and assuming

that Young’s modulus and density of the plate material vary with the space coordinates by the

functional relations

)1( 210 YXEE and )1( 210 YX .

Now, equation (6) reduces to

1

0

1

0

2

21

22

0

2224

1

0

1

0

22*2

0

2

)1()1(12

])1(22[

dYdXWYXa

dYdXWWWWWHE XYYYYYXXXX

(7)

where )1( 21

* YX and ba / .

The nondimensional bending moments ( ),, XYYX MMM and shear forces ( YX QQ , ) from Szilard

(Szilard 1974) are given by

,/),,(),,( 2ammmMMM XYYXXYYX aqqQQ YXYX /),(),( , (8)

where

))(1( 2

210 YYXXX WWYXDm

))(1( 2

210 XXYYY WWYXDm

XYXY WYXbaDm )1)(/( 210 ,

XWWYXDq YYXXX /)()1( 2

210 ,

YWWYXbaDq YYXXY /)()1)(/( 2

210 and )1(12/ 23

00 HED .



3 METHOD OF SOLUTION

In Rayleigh-Ritz method, the shape function should satisfy at least the essential boundary

conditions (Szilard 1974; Eschenauer et al. 1997; Shames and Dym 1985). This relaxation on

the shape function is due to the fact that natural boundary conditions that need not be satisfied

are implicitly contained in the functional as reported by Bathe(Bathe 1982) i.e. the kinematic

boundary conditions are automatically satisfied. Further, in a paper Wang et al. (Wang et al.

2002) reported that in case of commonly used Ritz method the stress resultants particularly

the twisting moments and shear forces are not generally predicted as accurately as the

deflection.

Now satisfying the essential boundary conditions, let us assume the displacement function

N

k

kk YXdYXW1

),,(ˆ),( (9)

where N is the order of approximation to get the desired accuracy, kd ’s are unknowns and

k̂ are orthonormal polynomials which are generated using Gram-Schmidt process as follows:

Orthogonal polynomials k over the region 10,10 YX have been generated with the

help of linearly independent set of functions ,........,3,2,1, kllL kk with

4321 )1()1(

ppppYYXXl , ......),.........,,,,,,,,,1( 322322 YXYYXXYXYXYXlk ,

where 21,01 orp as the edge 0X is free, simply supported or clamped. Same

justification can be given to 2p , 3p and 4p for the edges 0,1 YX and 1Y .

1

1

11 ,,k

j

jkjkk LL (10)

.,,.........4,3,2,)1(,......,3,2,1,,

,Nkkj

L

jj

jk

kj

The inner product of the functions say, 1 and 2 is defined as

1

0

1

0

212121 ),(),()1(, dYdXYXYXYX , (11)

where )1( 21 YX is the weight function and the norm of the function 1 is given

by

21

1

0

1

0

2

1212

1

111 ),()1(,||||

dYdXYXYX (12)

98 R. Lal et al.


The normalization can be done by using

.ˆk

kk

(13)

Using expression (9) into equation (7) and minimization of the resulting expression for 2 w.

r. t. kd ’s leads to the standard eigenvalue problem

,,........,3,2,1,0)( 2

1

Njda kjk

N

k

jk

(14)

where

dYdXFaYY

k

YY

j

XY

k

XY

j

YY

j

XX

k

YY

k

XX

j

XX

k

XX

jjk )ˆˆˆˆ)1(2)ˆˆˆˆ(ˆˆ( 422

1

0

1

0

,

(15)

)1( 21 YXF , 2

0

222

02 )1(12

HE

a and

kjif

kjifjk

,0

,1

The integrals involved in equation (15) have been evaluated using the formula

)!1()!1(

!!!!)1()1(

4321

4321

1

0

1

0

4321

pppp

ppppdYdXYYXX

pppp.

4 BOUNDARY CONDITIONS

The four boundary conditions namely CCCC, SCSC, FCFC and FSFS have been considered

in which C stands for clamped edge, S for simply supported edge and F for free edge. The

edge conditions are taken in anti-clockwise direction starting at the edge 0x (Figure 1(b))

and obtained by assigning various values to 321 ,, ppp and 4p as 0, 1, 2 for free, simply

supported and clamped edge conditions, respectively.

5 RESULTS AND DISCUSSION

The numerical values of the frequency parameter have been obtained by solving equation

(14) employing Jacobi method. The lowest three eigenvalues have been reported as the first

three natural frequencies corresponding to different boundary conditions considered here. The

values of various plate parameters for these three modes of vibration are taken as follows.

Nonhomogeneity parameters:

5.0,4.0,3.0,2.0,1.0,0.0,1.0,2.0,3.0,4.0,5.0,,, 2121 ,

aspect ratio: 00.2,75.1,50.1,25.1,00.1,75.0,50.0,25.0/ ba and υ=0.3.



To choose the appropriate value of the order of approximation N , a computer program

developed for the evaluation of frequency parameter was run for different values of N .

The numerical values showed a consistent improvement with the increasing value of N for

different sets of the values of plate parameters. In all the above computations, N =37 has been

fixed, since further increase in the value of N does not improve the results even at fourth

place of decimal. Table 1 shows the convergence of frequency parameter with N for a

particular set of plate parameters where maximum value of N was required.

Table 1: Convergence of frequency parameter with N for

5.0,5.0,5.0,5.0,1/ 21,21 ba .

The results are presented in table 2 and figures 2 to 5. It is observed that the frequency

parameter decreases in the order of boundary conditions CCCC, SCSC, FCFC and FSFS

for the same set of values of plate parameters. Figure 2 shows the effect of nonhomogeneity

parameter 1 on the frequency parameter for 5.02 , 5.01 , 5.02 and

1/ ba for the first two modes of vibration. It is observed that the frequency parameter

increases with increasing values of 1 for all the boundary conditions keeping other plate

parameters fixed. The value of decreases with the increasing values of 1 while increases

with the increasing value of 2 for fixed values of other plate parameters. The effect of 2

and 1 is more pronounced for 5.01 as compared to 5.01 . The rate of increase in

with 1 is in the order of the boundary conditions CCCC>SCSC>FSFS>FCFC when 1

changes from -0.5 to 0.5 and order becomes CCCC>SCSC>FCFC>FSFS when 2 changes

from -0.5 to 0.5, other parameters being fixed. This rate is higher in the second mode as

compared to the first mode.

CCCC SCSC

mode mode N I II III N I II III

10 35.1857 72.2623 72.9172 10 28.3376 54.5962 68.8044

20 35.1758 72.0477 72.6848 20 28.3302 54.2461 68.4158

30 35.1734 72.0318 72.6734 30 28.3296 54.2292 68.4019

35 35.1734 72.0306 72.6703 35 28.3296 54.2286 68.4010

36 35.1734 72.0304 72.6703 36 28.3295 54.2284 68.4010

37 35.1734 72.0304 72.6703 37 28.3295 54.2284 68.4010

FCFC FSFS

mode mode N I II III N I II III

10 21.6325 26.0597 47.0359 10 9.5491 16.3924 40.6461

20 21.5897 25.8879 43.7960 20 9.4956 16.2386 37.2695

30 21.5699 25.8553 43.4221 30 9.4943 16.2333 36.9152

35 21.5644 25.8302 43.4051 35 9.4940 16.2313 36.9079

36 21.5642 25.8296 43.4045 36 9.4940 16.2312 36.9079

37 21.5642 25.8296 43.4045 37 9.4940 16.2312 36.9079

100 R. Lal et al.


Figure 2: Frequency parameter for (a) CCCC, ( b) SCSC, (c) FCFC and (d) FSFS plate: for

.5.02 , first mode ; , second mode;

□, 5.02 , 5.01 ; o, 5.02 , 5.01 ; , 5.02 , 5.01 ; ×, 5.02 , 5.01 .

20

40

60

80

100

120

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

1

(a)

15

25

35

45

55

65

75

85

95

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

1

(b)

10

15

20

25

30

35

40

45

50

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

1

(c)

5

10

15

20

25

30

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

1

(d)



Figure 3: Frequency parameter for (a) CCCC, (b) SCSC, (c) FCFC and (d) FSFS plate: for

.5.02 , first mode; , second mode;

□, 5.02 , 5.01 ; o, 5.02 , 5.01 ; , 5.02 , 5.01 ; ×, 5.02 , 5.01 .

15

25

35

45

55

65

75

85

95

105

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

1

15

25

35

45

55

65

75

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

1

(b)

10

15

20

25

30

35

40

45

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

1

(c)

5

7

9

11

13

15

17

19

21

23

25

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

1

(d)

(a)

102 R. Lal et al.


Figure 4: Frequency parameter Ω for (a) CCCC, (b) SCSC, (c) FCFC, and (d) FSFS plate:

for 5.011 . , first mode ; , second mode ;

□, 5.0,5.0 22 ; o, 5.0,5.0 22 ; , 5.0,5.0 22 ;×, 5.0,5.0 22 .

10

30

50

70

90

110

130

150

0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

ba /

(a)

5

25

45

65

85

105

125

145

0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

ba /

0

10

20

30

40

50

60

70

80

90

100

0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

ba /

(c)

0

10

20

30

40

50

60

0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

ba /

(d)

(b)



Figure 3 depicts the behavior of the frequency parameter with the density parameter 1

for 5.01 , 5.02 , 5.02 and 1/ ba for the first two modes of vibration. It is

seen that the frequency parameter decreases with increasing value of 1 whatever be the

values of other plate parameters. The frequency parameter is found to increase with the

increasing value of 1 keeping other plate parameters fixed. The value of further

decreases with the increasing value of 2 . The rate of decrease in with 1 is in the order

of the boundary conditions CCCC>SCSC>FSFS>FCFC when 1 changes from -0.5 to 0.5

and order becomes CCCC>SCSC>FCFC>FSFS when 2 changes from -0.5 to 0.5. This rate

is more pronounced in the second mode as compared to the first mode.

(a) (b)

(c ) (d)

Figure 5: First three mode shapes of (a) CCCC, (b) SCSC, (c) FCFC and (d) FSFS square

plates for 5.011 and 5.022 .

The effect of aspect ratio ba / on the frequency parameter for 5.01 , 5.01 ,

5.02 , 5.02 for the first two modes of vibration has been shown in Figure 4. It is

clear that frequency parameter increases with the increasing value of ba / for all the

boundary conditions, other plate parameters being fixed. The values of is found to

104 R. Lal et al.


decrease with the increasing value of 2 and increases with the increasing value of 2 . The

rate of increase in with ba / is in the order of the boundary conditions

FCFC>SCSC>CCCC>FSFS when 2 changes from -0.5 to 0.5 and becomes

SCSC>CCCC>FCFC>FSFS when 2 changes from -0.5 to 0.5, keeping all other parameters

fixed. This rate of increase is much higher for ba / >1 as compared to ba / <1 and increases

with the increase in the number of modes.

Table 2: Frequency parameter Ω for CCCC, SCSC, FCFC and FSFS plates

CCCC SCSC

α1 -0.5 0.5 -0.5 0.5

α2 -0.5 0.5 -0.5 0.5 -0.5 0.5 -0.5 0.5

β1 β2 Mode

I 35.1734 49.9094 49.9095 60.8179 28.3295 40.2317 39.9856 48.8453

-0.5 II 72.0304 100.6960 100.6960 123.8370 54.2284 76.3725 77.7960 94.1107

III 72.6703 103.6270 103.6300 124.6980 68.4010 97.3166 93.9012 115.7170

-0.5

I 23.9259 35.8124 35.1365 43.7590 19.3280 28.8186 28.2558 35.1568

0.5 II 46.7713 73.1517 70.9388 88.8663 36.0189 54.6545 53.7848 67.0526

III 50.2592 73.2329 72.1782 89.7660 46.3214 69.1453 67.0006 83.5728

I 23.9259 35.1365 35.8124 43.759 19.1896 28.2558 28.8186 35.2298

-0.5 II 46.7715 70.9379 73.1519 88.8666 36.793 53.7849 54.6543 66.6411

III 50.2591 72.1774 73.2331 89.7674 44.4278 67.0009 69.1453 84.6306

0.5

I 19.2656 28.9911 28.9922 35.9133 15.4997 23.3017 23.3513 28.8934

0.5 II 37.4047 58.6728 58.6731 73.2942 29.2833 44.3506 44.0708 54.7060

III 40.4493 59.5342 59.5354 73.3265 36.2132 55.2903 56.0095 69.2490

FCFC FSFS

α1 -0.5 0.5 -0.5 0.5

α2 -0.5 0.5 -0.5 0.5 -0.5 0.5 -0.5 0.5

β1 β2 Mode

I 21.5642 30.3600 27.1953 34.0244 9.4940 13.4368 12.8591 15.8886

-0.5 II 25.8296 38.5664 42.1345 50.6625 16.2312 23.4462 24.4448 29.5922

III 43.4045 62.0228 64.6205 77.9202 36.9079 51.8983 48.1611 60.3256

-0.5

I 13.8135 22.0894 20.1626 25.5857 6.4992 9.6043 9.2914 11.5639

0.5 II 17.7829 26.3917 27.7833 34.0491 10.7955 16.1765 16.3209 20.1421

III 28.6390 43.6253 43.4448 53.9228 23.8478 36.7846 35.5626 45.0577

I 12.2048 20.1666 22.0894 26.9607 6.1672 9.2914 9.6043 11.7494

-0.5 II 19.8095 27.8014 26.3914 32.5144 11.3743 16.3210 16.1765 19.8346

III 29.6443 43.4460 43.6252 53.3246 21.7721 35.5692 36.7840 44.8460

0.5

I 9.9982 16.8951 17.8443 22.1529 5.0473 7.6751 7.8019 9.6197

0.5 II 15.6906 22.4377 21.3311 26.4538 9.0497 13.2638 13.0538 16.1541

III 23.5677 35.5716 35.1788 43.6298 17.8401 29.7411 29.6048 36.7530



0

0.2

0.4

0.6

0.8

1

00.10.20.30.40.50.60.70.80.91

-1

-0.5

0

0.5

1

y-axisx-axis (a)

0

0.2

0.4

0.6

0.8

1

00.10.20.30.40.50.60.70.80.91

-1

-0.5

0

0.5

1

y-axisx-axis

(b)

0

0.2

0.4

0.6

0.8

1

00.1

0.20.30.4

0.50.60.7

0.80.91

-1

-0.5

0

0.5

y-axisx-axis (c)

0

0.2

0.4

0.6

0.8

1

00.10.20.30.40.50.60.70.80.91

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

y-axisx-axis (d)

0

0.2

0.4

0.6

0.8

1

00.1

0.20.30.4

0.50.60.7

0.80.91

-1

-0.5

0

0.5

y-axisx-axis (e)

Figure 6: Normalized (a) XM (b) YM (c) XYM (d) XQ and (e) YQ of FCFC square plate for

0,5.0,5.0 2121 .

Table 3: Comparison of frequency parameter Ω for homogeneous

( 0,0,0,0 2121 ) square )1/( ba plate for 3.0 .

Boundary

Conditions

Ref. Mode

I II III

CCCC Leissa (1973) 35.992 73.413 73.413

Dickinson and Li (1982) 35.988 73.406 73.406

Bhat (1985) 35.986 73.395 73.395

Bhat et al. (1990) 35.986 73.395 73.395

Liew et al. (1990) 35.99 73.41 -

Classen and Thorne (1960) 35.985 73.394 -

Durvasula (1969) 35.991 73.413 -

Narita (2000) 35.99 73.39 -

Gutierrez et al. (1995) 36.01 - -

Present 35.9855 73.3954 73.3954

SCSC Leissa (1973) 28.951 54.743 69.327

Bhat et al. (1990) 28.951 54.743 69.327

Negm and Armanios (1983) 28.900 - -

Wang and Wang (2008) 28.9551 54.7466 69.3393

Laura et al. (1991) 28.95 54.88 69.34

Gutierrez and Laura (1995) 28.96 - -

Present 28.9509 54.7431 69.3270

FCFC Mizusawa (1986) 22.03 26.05 43.20

Leissa (1973) 22.27 26.53 43.66

Classen and Thorne (1960) 22.17 26.40 43.6

Present 22.1922 26.4510 43.6205

FSFS Mizusawa (1986) 9.631 16.13 36.73

Leissa (1973) 9.6314 - -

Bert and Malik (1996) 9.631385 16.134778 36.725643

Present 9.6314 16.1348 36.7256

106 R. Lal et al.


No special feature was observed from the graphs for third mode of vibration (figures not

given here) except that the rate of increase/decrease in the frequency parameter with a

specific parameter is higher than that for second mode. Three dimensional mode shapes for

specific plate have been shown in Figure 5. On the suggestion of the learned reviewer, the

bending moments ( )/(),,(),, 0DMMMMMM XYYXXYYX and shear forces

)/(),(),( 0DQQQQ YXYX for FCFC boundary condition have been plotted in figure 6.

A comparison of frequency parameter for homogeneous 02121 square

)1/( ba plate of uniform thickness with those obtained by other methods has been presented

in table 3. A close agreement of results is obtained.

6 CONCLUSIONS

The effect of non-homogeneity arising due to the dependence of Young’s modulus and

density of the plate material on both the variables x and y, on the natural frequencies of

isotropic rectangular plates of uniform thickness has been studied using boundary

characteristic orthogonal polynomials in Rayleigh-Ritz method on the basis of classical plate

theory. It is observed that the frequency parameter increases with the increasing values of

nonhomogeneity parameters 1 and 2 , aspect ratio ba / , while it decreases with increasing

values of density parameters 1 and 2 for all the four boundary conditions, keeping all other

plate parameters fixed. The percentage variation in the value of frequency parameter for

the first mode of vibration are -14.4 to 11.4, -14.2 to 11.4, -17.4 to 9.8 and -14.7 to 11.2 for

CCCC, SCSC, FCFC and FSFS boundary conditions, respectively when the nonhomogeneity

arises due to the change in only 1 from -0.5 to 0.5. The corresponding variations are -15.3 to

10.6, -15.2 to 10.6, -11.3 to 11.9 and -14.3 to 10.9 when 1 changes from -0.5 to 0.5.These

variations decrease by almost 0.5% with the increase in the number of modes. The present

analysis will be of great use to the design engineers dealing with nonhomogeneous plates in

obtaining the desired frequency by varying one or more plate parameters considered here.

ACKNOWLEDGEMENT

The authors wish to express their sincere thanks to Professor Bani Singh for his critical

discussions and useful suggestions during this work and the learned reviewers for their

constructive comments. One of the authors Yajuvindra Kumar is grateful to Council of

Scientific & Industrial Research, India for providing the senior research fellowship.

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